Geometry of Formal Varieties in Algebraic Topology

This talk has since been given to a live audience and typeset at 1 and 2.

< thermo> all right, it's noon by my clock :) the seminar topic is on one role
of algebraic geometry in algebraic topology. i've tried to make the vocabulary
in the seminar fairly unassuming; we'll see if i've succeeded
-!- ChanServ changed the topic of #mathematics to: If you have a question, type
'!' and wait for me to cede the floor.
< thermo> the impetus of algebraic geometry is to try to assign geometric
meaning to concepts coming from ring theory. the classical example is that,
given a set X of n-variate polynomials over a(n algebraically closed) field k,
we can associate the simultaneous vanishing locus of all elements of the set,
Z(X), in n-dimensional k-space. Z(X) is called an algebraic set, and
irreducible (in a sense) algebraic sets are called varieties
< thermo> a key thing to notice here is that if you have a larger X, you have
more polynomials, and so it gets harder for all of them to have zeroes in the
same spots --- and so the associated vanishing locus gets smaller. this means
that, if we expect any kind of correspondence of this type between algebra and
geometry, it has to be 'direction reversing'
< thermo> a lot of energy was invested in figuring out what setting we can build
that 1) supports all rings and 2) has nice geometric properties. varieties, for
instance, are insufficient; they turn out only to give a geometric description
of finitely generated k-algebras with no nilpotent elements
< thermo> eventually, things called schemes were invented and people presented
them as locally ringed spaces --- this is what everyone sees in a first or
second semester algebraic geometry course. since i come from topology, i find a
different presentation of schemes much easier to think about: their functors of
points
< thermo> from a categorical perspective, this presentation of the opposite
category of rings is almost obvious; to every ring R we can build an object
Hom(R, -), which is functor that assigns rings S to the set of ring
homomorphisms R --> S. the assignment spec: R |-> Hom(R, -) is contravariant
(i.e., inclusion-reversing), and so provides an embedding of Rings^op into some
larger category of functors off the category of rings, which I
< thermo> will call the category of schemes
< thermo> hmm have to keep an eye on long lines
< thermo> while not very obviously geometric, this idea of studying rings by
studying the maps off them turns out to be a very good idea. in a sense, rather
than asking "what is the ring R?" it suggests that we instead ask "what does the
ring **DO**?"
< thermo> to illustrate what i mean, take the polynomial ring Z[x], and
associate to it the scheme spec Z[x]. we can apply spec Z[x] to a test ring S
and see what we get: (spec Z[x])(S) = hom(Z[x], S). a map Z[x] --> S is
determined completely by where x is sent and x can be sent to any element --- so
spec Z[x] models the forgetful functor Rings --> Sets. this is kinda neat
< thermo> to get a sense of what else rings can do, recall that each ring R
comes with an additive group (R, +) on its underlying set of points, and maps
between rings give rise to maps between these additive groups
< thermo> the Yoneda lemma is a fundamental result of category theory, which in
one language says that structure visible on the image of a representable functor
actually lives on the representing object --- so we should expect a group
structure on the scheme spec Z[x]
< thermo> or, since spec is direction-reversing, we could call this a cogroup
(or 'Hopf algebra') structure on the ring Z[x]. to illustrate, we should get a
map Z[t] --> Z[x] (x) Z[y] corresponding to the 'opposite' of addition
< thermo> in terms of the action of spec, to any two maps Z[x] --> S and Z[y]
--> S (called S-valued points of the schemes spec Z[x] and spec Z[y]) we need to
associate a map spec Z[t] --> S corresponding to their sum
< thermo> the content of Yoneda's lemma is that it's enough to do this "in the
universal case": the identity maps Z[x] --> Z[x] and Z[y] --> Z[y] classify the
elements x and y, and hence Z[t] --> Z[x] (x) Z[y] should classify the element x
+ y --- and this is the opposite addition map we wanted
< thermo> this gives rise to a comultiplication map on Z[x], and so a
'multiplication' map on spec Z[x]. spec Z[x] together with this group structure
we call the 'additive group (scheme)', written Ga
< thermo> this happens *all the time* with all kinds of interesting functors on
the category of rings
< thermo> another functor everyone knows is the group of units: there's a
functor Gm with Gm(R) equal to the invertible points of R. to get a point of
Gm, we need to select an element x in R and another element y playing the role
of its inverse, so xy = 1. using this logic, we can model Gm as spec Z[x, y] /
<xy - 1>. this scheme is called the 'multiplicative group'
< thermo> shifting gears slightly, one of the classical presentations of
infinitessimal elements are elements that 'square to zero'. this turns out to
be a fairly difficult thing to make both precise and analytically useful, which
is why most analysts spend their time thinking about other things --- but it's a
very useful notion in algebraic geometry. we think of the nilpotent elements of
a ring R as being infinitessimal
< thermo> there's a very useful category related to our category of schemes.
let R be a topologized augmented k-algebra whose augmentation ideal I is
topologically nilpotent --- the category of such objects is called the category
of adic k-algebras
< thermo> (with maps continuous homomorphisms)
< thermo> we can do the same functor-of-points construction here, but the
resulting schemes are called 'formal schemes' --- they're contravariant functors
from adic k-algebras to Sets.
< Kasadkad> !
< thermo> yeah?
< Kasadkad> what's topologically nilpotent?
< thermo> it means that for any n there's an m so that x^m lies in I^n, where R
has the I-adic topology
< thermo> so rather than requiring actual nilpotence, you mean the limit of the
powers of the elements tends to zero in this sense
< thermo> that make sense?
< Kasadkad> yes
< thermo> cool
< Kasadkad> and the augmented algebra bit means R/I =~ k?
< thermo> yeah
< Kasadkad> ok
< thermo> so, as an example, the ring of power series k[[x]] is an augmented
adic k-algebra with the <x>-adic topology, and so gives a formal scheme spf
k[[x]]. this example is particularly relevant --- there's a subcategory of
formal schemes called formal varieties, consisting of those functors which are
(noncanonically) isomorphic to spf k[[x1, ..., xn]] for some n
< thermo> the name, for those interested, comes from taking the sheaf of
functions of a smooth algebraic variety and completing at a point. what you get
is a k-algebra of this type, where the number of indeterminates is equal to the
dimension of the variety
< thermo> and this is how you should think of a formal variety. just like a
smooth manifold has about every point a neighborhood where it looks like
euclidean space, a smooth variety has about every point an infinitessimal
neighborhood where it looks like 'formal n-space', and we're interested in that
tiny neighborhood
< thermo> some more vocabulary: a formal variety V is something noncanonically
isomorphic to spf k[[x1, ..., xn]] = A^n. a selected isomorphism between V and
A^n is called a 'coordinate' on V, corresponding to picking charts on a
manifold. the space of maps A^n --> A^m corresponds to m-tuples of n-variate
power series, just like maps between charts on analytic manifolds
< thermo> and now here i thought i'd take a break to let you catch your breath
and ask questions if you have any
< Kasadkad> "spf k[[x1, ..., xn]] = A^n" is just introducing the notation "A^n"?
< thermo> yeah
< thermo> and 'spf', which i think i forgot to define, is the analogue of 'spec'
for formal schemes
< Kasadkad> so i gathered
< thermo> ok, let's get back to it: let's apply some of these words to algebraic
topology. in first or second semester algebraic topology, one thing everyone
learns about singular cohomology is that it supports a ring structure ---
there's a notion of 'cup product' H^n M x H^m M --> H^(n+m) M for a space M
< thermo> as budding algebraic geometers, rather than trying to write down what
these rings are, we know that we should try to write down what these rings 'do'.
< thermo> this might not sit well with those of you that haven't seen lots of
algebraic topology; the examples given in an introductory course have very
sparse information, but basically all spaces of lasting interest to a topologist
have extremely complicated cohomological information embedded
< thermo> the cohomology ring associated to S^n, for instance, isn't so gripping
--- but K(Z/p, n), for instance, is
< thermo> at any rate, this suggests that rather than thinking of H^* M as a
cohomology ring, we should think of M_H = spf H^* M as a formal scheme (where
the ideal inducing the I-adic topology corresponds to cohomological information
above degree 0) over the formal scheme S_H, detecting the coefficient ring of H
< thermo> (in fact, H can be replaced by any ring-valued cohomology theory to
produce a similar theory of schemes associated to spaces)
< thermo> (one thing the reader should note (and object to!) is that this
construction is extremely insensitive to issues with grading. namely, we've
thrown away the natural grading given to us by cohomological dimension, and
we've even assumed that the odd-dimensional parts vanish, so that we get a
strictly commutative ring. this is a lot to ask, and introduces a few problems
that we'll ignore :) )
< thermo> so, what are some examples where this ideology is a good idea in
topology? one calculation that everyone should know is the singular cohomology
of infinite dimensional complex projective space: H^* CP^infty = Z[[x]], with x
a cohomology class in degree 2
< thermo> (those of you who are especially picky might also object here to the
use of power series rather than a polynomial ring; this is totally a matter of
taste in taking a limit or a colimit when building the cohomology ring from its
graded pieces, and i'm going to take a limit because it makes my theory much
more interesting)
< thermo> so, CP^infty_H = spf H^* CP^infty is isomorphic formal affine line,
A^1 = spf Z[[x]], where our choice of x in the cohomology of CP^infty gave us a
coordinate on CP^infty_H. that's neat, albeit not too interesting; it's the
most basic formal scheme we know of, pretty much
< thermo> to help make it interesting, there is a group structure on CP^infty,
as the classifying space for line bundles, corresponding to the universal tensor
product of line bundles
< thermo> let's study what structure this induces on the formal scheme
associated to CP^infty by computing the product map CP^infty_H x CP^infty_H -->
CP^infty_H using the coordinate we have
< thermo> we effectively need to ask what x pulls back to under the
multiplication map. well, a priori, we have that it pulls back to some
bivariate power series F in H^* (CP^infty x CP^infty) = Z[[x, y]]
< thermo> the niceness of tensor product tells us a few things about F: the
tensor product is commutative, so F(x, y) = F(y, x). the tensor product
satisfies L (x) 1 = L for any bundle L, so F(x, 0) = x. lastly, the tensor
product is associative, (L (x) L) (x) L = L (x) (L (x) L), so we
< thermo> in ordinary cohomology, we can actually calculate F. the Kunneth
formula says that in the calculation H^*(CP^infty x CP^infty) = Z[[x, y]] with x
and y both of degree 2, so the only possible thing F could be is F(x, y) = x +
y, due to degree restrictions
< thermo> --- but this is already interesting. :) another interpretation of this
calculation is that the induced map CP^infty_H x CP^infty_H --> CP^infty_H
corresponding to the product on formal schemes is given by the power series F(x,
y) = x + y, which, as we calculated earlier, is exactly the function
corresponding to the addition in the additive group
< thermo> and so CP^infty_H is another name for Ga
< thermo> let's do the same calculation for K-theory, another kind of cohomology
theory, where K^* X corresponds basically to vector bundles over X. for some
topological reasons, writing R for the coefficient ring of K-theory, K^*
CP^infty also turns out to be R[[x]], and so CP^infty_K is also an affine line
< thermo> CP^infty_K comes with a natural choice of coordinate: namely, to the
universal line bundle L over CP^infty, we associate the K-theoretic class [L]-1
corresponding to L (the -1 is to fix some dimensional issues; you should not
think too hard about it)
< thermo> that tensor product corresponds to multiplication in K-theory means
that we can calculate the action of the tensor product manually:
< thermo> this is kind of ugly over IRC :) but: write c(L) for the image of x
under pullback along some map M --> CP^infty classifying a line bundle L over M.
then c(L (x) L') = [L][L'] - 1 = [L][L'] - [L] - [L'] + 1 + [L] - 1 + [L'] - 1 =
[L]-1)([L']-1) + [L]-1 + [L']-1 = c(L)c(L') + c(L) + c(L'), and so F(x, y) = xy
+ x + y
< thermo> here's the real punchline, so you can skip the previous line: if you
work it out, this is exactly the product structure associated to the
multiplicative group Gm with the coordinate x - 1 !
< thermo> so CP^infty_K is isomorphic to Gm
< thermo> hmm looks like i sped up a bit again
< Kasadkad> i'm with you
< thermo> cool :)
< thermo> briefly: some other less pivotal examples include BU(n)_H, which is
isomorphic to n-dimensional affine space, with group structure corresponding to
divisors of degree n on S_H. in the limit, BU_H corresponds to the formal group
of divisors on S_H
< thermo> at a prime p, the even part of spec H_* K(Z, 3) turns out to
corepresent the functor of Weil pairings Ga^2 --> Gm
< thermo> if you're familiar with Morava K-theory, K(Z, *)_K(n) is the free ring
object on BZ/p^infty_K(n), which is also tied in with Weil pairings on
p-divisible groups
< thermo> i don't really want to explain either of these examples, i just want
to point out that this geometric language organizes huge calculations elsewhere
in topology too
< Kasadkad> !
< thermo> sure
< Kasadkad> what is S_H, at least?
< thermo> oh, did i not explain that earlier? S_H is H^* of a sphere, so spf of
the coefficient ring of H
< Kasadkad> don't think so, ok
< thermo> it corresponds to the 'k' in 'category of adic k-algebras'
< thermo> since every based space X comes with maps pt --> X --> pt, you get an
augmented H^* pt-algebra structure on H^* X by H^* pt <-- H^* X <-- H^* pt
< thermo> gonna take another couple minute break
< thermo> all right, back to it:
< thermo> let's stick to analyzing CP^infty, since we recovered our two earlier
examples with it and there seemed to be a lot left, as those axioms we imposed
on F were much laxer than "F is x + y OR F is x + y + xy."
< thermo> the cases where our analysis of the situation thus far goes through is
when CP^infty_E is isomorphic to the affine line over S_E for our cohomology
theory E of interest. these cases will be called 'complex orientable', and a
choice of coordinate on CP^infty_E is a 'complex orientation'. the resulting
bivariate power series F is called a formal group law
< thermo> the absolute most important theorem in the study of complex oriented
spectra has to do with the structure of a cohomology theory called complex
cobordism, MU. the theorem, due to Quillen, has two parts, i'll talk about both
in turn, but not about their proofs, which involve a complicated piece of
topology called H_infty ring spectra and generalized power operations
< thermo> the first thing we should notice is that the space of formal group
laws itself constitutes a scheme
< thermo> there is a scheme representing all bivariate power series: spec
Z[a_00, a_10, a_01, a_20, a_11, a_02, ...] = spec P, which supports the power
series f = sum_{i, j} a_ij x^i y^j. any other power series over any other ring
R occurs uniquely as a pullback of f along a map spec R --> spec P (equiv, a map
P --> R) selecting the coefficients of the power series
< thermo> then, we can impose the conditions on the representing ring specified
by the axioms of formal group laws through a quotient, since those are all
algebraic --- just like we did for Gm = spec Z[x, y] / <xy - 1>
< thermo> symmetry, for instance, asserts that a_{ij} - a_{ji} = 0, and rigidity
states that a_{i0} = 0 for all i >= 1, and the associativity condition is much
harder to write down. the resulting ring is called L, since Lazard first
studied it and found that it was (noncanonically) isomorphic to the infinite
polynomial ring Z[c2, c3, ...] --- a very nontrivial result!
< thermo> the first part of Quillen's first theorem states that S_MU, the
coefficient ring of MU, is (noncanonically) isomorphic to spec L
< thermo> the second part says that this is really a topological statement: if E
is a complex oriented ring spectrum and we've chosen an orientation for MU, then
there is a unique map multiplicative map MU --> E of cohomology theories (in
fact, of spectra) so that S_E --> S_MU = spec L classifies the formal group law
associated to the complex orientation of E
< thermo> this is pretty nuts on its own --- this means that the entire theory
of commutative one-dimensional formal group laws embeds faithfully into
algebraic topology through CP^infty and the tensor product of complex line
bundles. this has a *lot* to say about the algebraic geometry associated to
cohomology theories!
< thermo> but it's not over yet! the product spectrum MU ^ MU comes with two
maps from MU: there's inclusion on the left and inclusion on the right
< thermo> correspondingly, we have two complex orientations of CP^infty_{MU ^
MU}, but we the scheme CP^infty_{MU ^ MU} exists totally in isolation of the
words 'complex orientation' --- so what we're really saying is that we have two
coordinates of CP^infty_{MU ^ MU}, and they are necessarily related by an
invertible change of coordinates
< thermo> for almost formal reasons, this turns out to mean that, just like the
coefficient ring of MU carried the universal example of a formal group law, the
coefficient ring MU ^ MU carries the universal example of a formal group law
*isomorphism*. this is quillen's theorem pt 2
< thermo> in topological terms, this means that the homology cooperations
associated to MU correspond to reparametrizations of its formal group. to
explain, to all cohomology theories E we get a map E^* E (x) E^* X --> E^* X
corresponding to an action of 'cohomology operations', and in nice cases we get
a reverse map in homology E_* X --> E_* X (x) E_* E corresponding to a coaction
of 'homology cooperations'
< thermo> geometrically, what does this mean? Lazard's ring L corresponds to
the 'moduli space of formal group laws', in the sense that to any formal group
law F over any ring R, we get a map spec R --> spec L pulling back the universal
formal group law to F. but, using our language of formal varieties, we see that
in doing so we've made an arbitrary choice --- what we'd really like is a formal
group, without a choice of coordinate
< thermo> gosh what a sea of text
< thermo> correspondingly, the group of automorphisms of A^1 as a formal variety
act on spec L, corresponding to the action of S_{MU ^ MU} on S_MU. over an
arbitrary formal group spec R with a coordinate and formal group law F, we have
a slightly smaller action: Aut F acts on spec R
< thermo> this data assembles into what is called a(n affine, algebraic) stack.
in the language of functors of points, what we're saying is that we're
associating more than just a set of formal group laws to R: we're also
associating a bunch of isomorphisms between them, so that taking connected
components of the resulting groupoid gives you the space of formal group
structures on spec R
< thermo> in fact, the interpretation of the coefficient ring of MU ^ MU as
reparametrizations of the universal formal group law and as the homology
operations associated to complex bordism means that the homology cooperation
coaction on MU_* X means that MU_* X gives rise to a automorphism-equivariant
sheaf on S_MU, or equivalently a sheaf on this stack
< thermo> --- well, before that means anything, i guess i should say what a
(quasicoherent) sheaf of modules is over a formal scheme X
< thermo> to each ring R and map spec x: R --> X (called an R-valued point of
X), we assign an R-module M_x in a functorial way (i.e., maps spec S --> spec R
--> spec X push modules forward up to isomorphism and so forth).
< thermo> whoops x: spec R --> X, and spec S --> spec R --> X
< thermo> in our setting, this is really not an exciting statement --- MU_* X is
indeed an MU_*-module, and any map spec R --> S_MU coming from a map f: MU_* -->
R gives rise to an R-module: R (x)_{MU_*} MU_* X. this is all we meant. :)
< thermo> what *is* exciting is the existence of the Adams spectral sequence,
which is some functorially available spectral sequence which converges to the
homotopy of a space X. the E_2 page in topologist's language starts with
Ext_{*, *}^{MU_* MU}(MU_*, MU_* X), which in geometer's language is the
cohomology of the sheaf associated to MU_* X over the stack of formal groups
< thermo> what this is indicating is that the category of quasicoherent sheaves
of modules over the moduli stack of formal groups is **very** similar to the
stable homotopy category, and the geometry of sheaves can tell us a whole lot
about the global structure of the entire field of stable homotopy
< Kasadkad> !
< thermo> go ahead
< Kasadkad> what does MU_* MU (or E_* E) mean
< thermo> E_* E is shorthand for the homotopy groups of the product E ^ E
< Kasadkad> hmm
< thermo> , equivalently, the coefficient ring of the theory E ^ E
< Kasadkad> not sure what the product E ^ E is
< thermo> let me paste two more things, then i have another break written in
before we make everything concrete, so i'll go back and sketch out what you
should expect these things to look like. this question really boils down to
'what is a generalized cohomology theory'
< Kasadkad> k
< thermo> here are my two things:
< thermo> probably the most striking example of this relation after complex
orientations themselves is gross-hopkins duality, which i don't fully understand
yet. the core of it is to use projective geometry to make strong statements
about what the moduli stack of formal groups looks like, out of which comes some
really striking topology. maybe someday i can say something useful :)
< thermo> and also one way that this picture could be improved is to stop
pushing down to homotopy/homology/cohomology groups everywhere, which is what
we've had to do in order to say 'ah, look, here's a stack!' derived algebraic
geometry is what you get when you try to build this picture without performing
this lossy step from homotopy theory to algebra
< thermo> ok, so, regarding what E ^ E is and so forth:
< thermo> so another thing you should know from basic cohomology theory is that
the cohomology groups H^n(M; G) of a space are representable as homotopy classes
of maps M --> K(G, n) for some special target spaces K(G, n), called
Eilenberg-Mac Lane spaces
< thermo> the key relation among these spaces is that if you build the loopspace
of K(G, n), you get something homotopy equivalent to K(G, n-1)
< thermo> this is the picture that generalizes well. a generalized cohomology
theory E is made up of a sequence of spaces (E_0, E_1, ...) with Loops E_n
homotopy equivalent to E_{n-1}
< thermo> and so an E-cohomology class of a space X corresponds to some homotopy
class X --> E_n
< thermo> the language of spectra is invented so that you can think of these
constituent spaces as one big collective --- rather than having a homotopy class
X --> E_n, you instead build a homotopy class of maps of spectra X --> E
< Kasadkad> mm
< thermo> which raises the question: what should a generalized homology theory
be?
< thermo> well, it turns out that these available maps E --> X themselves form a
spectrum, called the function spectrum F(X, E)
< thermo> and the function spectrum has that nice adjoint property that you've
come to expect from elsewhere in math --- there's some notion of a product on
the category of spectra, written ^, so that maps(X ^ Y, Z) is the same as
maps(X, F(Y, Z))
< Kasadkad> hmm is that coming from smash product or something
< thermo> and so as '^' is dual to 'F(-, -)', the right interpretation of
'E-cohomlogy is dual to E-homology' turns out to be to define E-homology groups
of X as the path components of the product spectrum E ^ X
< thermo> yes, it is very much a smash product, but it's tricker than just
taking smash products on the individual spaces E_n or whatever
< Kasadkad> ok
< thermo> it was a huge source of headaches in the 70s and 80s --- but that's
how you should think of it
< Kasadkad> yes i was thinking of it as a huge source of headaches already
< thermo> anyway, this smash product on the level of spectra also lets you ask
what the E-homology of a spectrum F is
< thermo> haha
< thermo> E_* F is defined to be pi_* E ^ F
< thermo> does that answer your original question? i forget what it even was
< Kasadkad> about E_* F and E ^ E
< Kasadkad> E ^ F w/e
< Kasadkad> and yes
< thermo> cool
< thermo> so there's one more leg of the seminar to go, where we actually take
these lofty ideas and compute some thangs
< thermo> let's restrict to thinking about k a field of positive characteristic
< thermo> to a formal group law F over k, we associate an integer n, called its
height, as follows: let [m]_F(x) denote x +_F ... m times ... +_F x. then it
turns out that [p]_F(x) is always of the form g(x^(p^n)) for some n >= 0 with
g'(0) nonzero, and the integer n is called the height of the formal group
< thermo> it is basically measuring the size of the kernel of the
multiplication-by-p map on the formal group
< thermo> a theorem of Cartier says that this invariant is exceptionally strong:
when k is a perfect field of positive characteristic, a formal group over k is
uniquely characterized by its height. (moreover, there is a canonical
coordinate, called the Cartier coordinate, associated to each formal group, that
is particularly 'nice' in certain senses. we won't need this part)
< thermo> so, what this means is that the quotient of the moduli stack of formal
group laws by the coordinate change group (i.e., the orbit space) turns into a
bunch of points: N u {infty}. we'd really like to understand the isotropy
groups of these individual pieces
< thermo> for instance, let's restrict to the substack of additive formal groups
over the prime 2. there's an obvious formal group law, which we've already
encountered: F(x, y) = x + y. we can ask: what's the isotropy group of F?
< thermo> well, that's the space of maps A^1 --> A^1 commuting with the group
structure on both sides, i.e., a power series f(x) with f(x + y) = f(x) + f(y)
< thermo> in characteristic 2, these are easy to classify: they're the power
series of the form f(x) = x + sum_{i>0} a_i x^{2^i} for some sequence of
coefficients a_i in the base field --- which is to say that f can be picked out
by the scheme spf Z[a_1, a_2, ...] = Aut Ga
< thermo> the group structure on the isotropy group comes from composition of
these power series: let g(x) = x + sum_{j>0} b_j x^{2^j}, then g(f(x)) = sum_j
b_j (sum_i a_i x^{2^i})^{2^j} = sum_n x^{2^n} (sum_{i+j = n} b_j a_i^{2^j})
< thermo> the corresponding map of formal schemes Aut Ga x Aut Ga --> Aut Ga
sends c_n to sum_{i+j = n} b_j a_i^{2^j} --- and this is exactly the milnor
diagonal on the dual of the steenrod algebra, which took topologists decades and
much more than 2 lines to unearth
< thermo> so the isotropy group of the formal group law associated to a complex
oriented cohomology theory E is somehow telling us about the homology
cooperations associated to E
< thermo> this shouldn't be a huge surprise --- since a complex orientation of E
corresponds to a map MU --> E, we also get an induced map MU ^ MU --> E ^ E, and
hence of homology cooperations. it is a very useful guide for guessing what E ^
E looks like in many, many cases
< thermo> (another note for the careful reader: the cooperation dual to the
bockstein for p > 2 is missing from this picture as i've sketched it. it can be
accounted for, but it takes some work.)
< thermo> what about the automorphisms above the multiplicative group law, F = x
+ y + xy --- the one associated to K-theory? well, first we need to work with
p-adically completed K-theory, since we've made assumptions about how the base
field looks and so forth. then, multiplication by an integer in the formal
group structure induces a map n |-> [n]_Gm(x) = (1+x)^n - 1, Z --> Aut_k(Gm)
< thermo> what's neat is that this map extends to a map from the p-adic integers
to Aut_k(Gm) by alpha |-> sum_{k >= 1} (alpha choose k) x^k. the units of this
ring turn out to exhaust the automorphisms of p-adically completed K-theory,
corresponding to the Adams operations
< thermo> i don't actually know much about this interpretation of the Adams
operations, so i won't say more
< thermo> there is an obvious question in another direction though: what if
rather than considering the map MU ^ MU --> E ^ E, we instead have two complex
oriented spectra, MU --> E and MU --> F? then we get a map MU ^ MU --> E ^ F.
what does this buy us?
< thermo> in many good cases, the coefficient ring of E ^ F looks like you'd
expect --- it's a ring that corepresents the isomorphisms from the formal group
law associated to E-theory to the one associated to F-theory, i.e., the space of
power series f with f(x +_E y) = f(x) +_F f(y)
< thermo> and f invertible
< thermo> another thing that's worth noting is that this actually sometimes
gives unstable information too! if we destabilize the spectra E and F into a
sequence of infinite loop spaces E_* and F_*, then the E-homology of the
sequence of spaces F_* forms what's called a Hopf ring, which is supposed to
corepresent the space of formal group **homomorphisms** CP^infty_E -->
CP^infty_F
< thermo> this is also a very fruitful idea that lets you compute, for instance,
the Hopf ring BP_* BP_*, or K(n)_* K(n)_*, or K(n)_* K(Z, *) (<-- this was
mentioned earlier). steve wilson, doug ravenel, and neil strickland spent a
good chunk of their lives computing huge, amazing things using this as a
guideline
< thermo> at any rate, all of this should be very exciting. cohomology theories
are immensely complicated objects, but it frequently turns out that their rings
of (co)operations --- also immensely complicated objects --- are controlled by
isotropies of formal group laws, which are really pretty tractable as these
things go
< thermo> stumbling across the milnor diagonal was really supposed to be the
punchline :) serre's work in the 60s spanned several papers, trying to compute
the steenrod algebra using exceptionally complicated (for the time) spectral
sequence techniques
< thermo> and i think i'm about done. anything leftover that i can clarify or
expand on?
< Kasadkad> I'm not sure what [m]_F(x) meant
< Kasadkad> or rather what +_F meant
< thermo> x +_F y is another notation for F(x, y). applying the group law F to
the elements x and y
< Kasadkad> ok that's what I figured
< thermo> so x +_F x is like 'two times x', sort of, and we write [2](x)
< thermo> gonna switch back to irssi now that i'm not pasting things it might
cut off
< thermoplyae> phew
< Kasadkad> wb
< thermoplyae> so, putting aside questions about the material for a moment, i
will probably have to give a similar talk in front of a Real Live Audience
inside of this coming month. they know some topology, but the smartest one
among them asked me whether it was worth his time to learn any algebraic
geometry at all --- so i figured i'd do something like this to tell him: **YES**
< thermoplyae> are there things that you think ought to be changed for a second
run of this?
< Kasadkad> hmm
< thermoplyae> the ending could be modified, so that i at least don't have to
say 'hey look you might have missed the punchline but that happened a couple
minutes ago'
< thermoplyae> but beyond that, idk
< Kasadkad> heh
< Kasadkad> i don't really know the material well enough to have suggestions for
presenting anything differently
< thermoplyae> but as it was presented, it didn't feel like too much was
omitted? you could actually follow point to point, and things were somewhat
justified
< Kasadkad> yeah
< thermoplyae> well that's something
< Kasadkad> i kept up for longer than i expected
< Kasadkad> and when my eyes were glazing over, i think it was on account of not
really knowing topology
< Kasadkad> so if they know some
< thermoplyae> that's what they tell me
< thermoplyae> they spent last week constructing the steenrod squares and intend
to continue this week on tuesday, so i guess this punchline will be especially
potent since it doesn't take 3 hours to state
< thermoplyae> it's a little encouraging about how much topology they actually
know though. idk
< Kasadkad> who are "they", anyway, some topology classes?
< Kasadkad> *-es
< thermoplyae> some people attending this student-run seminar
http://math.berkeley.edu/~aaron/xkcd/
< thermoplyae> grad students from berkeley and one guy from stanford
< burned> why would they call it that?
< Kasadkad> ah
< thermoplyae> burned: idk, bad decisions
< thermoplyae> at any rate, they're interested in topology and competent, but
they're young and maybe not experienced or learned
< burned> eeww
< burned> and nerds, so they are a triple threat
< jimi_> Where is the log going to be?
< thermoplyae> good question
< thermoplyae> yourwiki is still down, looks like